Ch. 7 Review
IB Statistics
1. Suppose X is a random variable with mean μ. Suppose we observe X many times and
keep track of the average of the observed values. The law of large numbers says that
A) the value of μ will get larger and larger as we observe X.
B) as we observe X more and more, this average and the value of μ will get larger and
larger.
C) this value will get closer and closer to μ as we observe X more and more often.
D) as we observe X more and more, this average will get to be a larger and larger
multiple of μ.
E) none of the above.
2. In a population of students, the number of calculators owned is a random variable X
with P(X = 0) = 0.2, P(X = 1) = 0.6, P(X = 2) = 0.2. The mean of this probability
distribution is
A) 0 B) 2 C) 1 D) 0.5 E) not enough information given.
3. Refer to the previous problem. The variance of this probability distribution is
A) 1 B) 0.63 C) 0.5 D) 0.4 E) not enough information given.
4. The number of calories in a one-ounce serving of a certain breakfast cereal is a random
variable with mean 110. The number of calories in a full cup of whole milk is a random
variable with mean 140. For breakfast you eat one ounce of the cereal with ½ cup of
whole milk. Let Z be the random variable that represents the total number of calories in
this breakfast. The mean of Z is
A) 110 B) 140 C) 180 D) 250 E) 195
5. The weight of reports produced in a certain department has a Normal distribution with
mean 60 g and standard deviation 12 g. What is the probability that the next report will
weigh less than 45 g?
A) 0.1042 B) 0.1056 C) 0.3944 D) 0.0418 E) not enough information given.
6. Let X and Y be discrete random variables and let a and b be constants. Which of the
following is FALSE?
A) mean (X + Y) = mean (X) + mean (Y)
B) mean (X – Y) = mean (X) – mean (Y)
C) mean (aX) = (a)(mean (X))
D) mean (a + bx) = a + b mean X
E) If X and Y are independent, then mean (X/Y) = mean (X)/mean (Y)
7. X and Y are independent random variables, and a and b are constants. Here are some
statements about variances and standard deviations.
I.
Var (X + Y) = Var (X) + Var (Y)
II.
X + Y = X+Y
III.
Var (a + bX) = b Var (X)
IV.
X – Y = X – Y
V.
Var (X – Y) = Var (X) + Var (Y)
Which of the following statements are TRUE?
A) V
B) I, V
C) I, II
D) III, V
E) none of the statements is true.
8. A randomly chosen subject arrives for a study of exercise and fitness. Consider these
statements.
I.
After 10 minutes on an exercise bicycle, you ask the subject to rate his or her
effort on the Rate of Perceived Exertion (RPE) scale. RPE ranges in wholenumber steps from 6 (no exertion at all) to 20 (maximum exertion).
II.
You measure VO2, the maximum volume of oxygen consumed per minute
during exercise. VO2 is generally between 2.5 liters per minute and 6 liters
per minute.
III.
You measure the maximum heart rate (beats per minute).
The statements that describe a discrete random variable are
A) none of the statements describes a discrete random variable.
B) I
C) II
D) I, III
E) I, II, III
Use the following to answer questions 9 and 10:
A psychologist studied the number of sudoku puzzles subjects were able to solve in a fifteenminute period while listening to classical music. Let X be the number of sudoku puzzles
completed successfully by a subject. The psychologist found that X had the following probability
distribution:
Value of X
Probability
1
0.2
2
0.4
3
0.3
4
0.1
9. Using the above data, what is the probability that a randomly chosen subject completes
at least three puzzles in the fifteen-minute period while listening to classical music?
A) 0.3 B) 0.4 C) 0.6 D) 0.9 E) not enough information given.
10. Using the above data, P(X < 3) is
A) 0.3 B) 0.4 C) 0.6 D) 0.9
E) not enough information given.
Use the following to answer questions 11-13:
Let the random variable X be a randomly generated number with the uniform probability density
curve given below.
11. Referring to the information above, P(X = 0.25) is
A) 0 B) 0.025 C) 0.25 D) 0.75 E) 1
12. Referring to the information above, P(X  0) has value
A) 0 B) 0.1 C) 0.5 D) 1
E) The value cannot be determined since X must be greater than 0.
13. Referring to the information above, P(0.7 < X < 1.1) has value
A) 0.3 B) 0.4 C) 0.6 D) 0.7 E) 1.1
14. Which of the following random variables should be considered continuous?
A) The time it takes for a randomly chosen woman to run 100 meters.
B) The number of brothers a randomly chosen person has.
C) The number of cars owned by a randomly chosen adult male.
D) The number of orders received by a mail-order company in a randomly chosen
week.
E) None of the above.
Use the following to answer questions 15-17: The probability density curve of a random variable
X is given in the figure below.
15. Referring to the information above, the probability that X is between 0.5 and 1.5 is
A) 1/4 B) 1/3 C) 1/2 D) 3/4 E) 1
16. Referring to the information above, the probability that X is at least 1.5 is
A) 0 B) 1/4 C) 1/3 D) 1/2 E) 2/3
17. Referring to the information above, the probability that X = 1.5 is
A) 0 B) 1/4 C) 1/3 D) 1/2 E) 3/4
18. A random variable is
A) a hypothetical list of the possible outcomes of a random phenomenon.
B) any phenomenon in which outcomes are equally likely.
C) any number that changes in a predictable way in the long run.
D) a variable used to represent the outcome of a random phenomenon.
E) a variable whose value is a numerical outcome associated with a random
phenomenon.
19. Let the random variable X represent the profit made on a randomly selected day by a
certain store. Assume that X is Normal with mean $360 and standard deviation $50.
What is the value of P(X > $400)?
A) 0.2119 B) 0.2881 C) 0.7881 D) 0.845 E) 0.9999
20. A business evaluates a proposed venture as follows. It stands to make a profit of $10000
with probability 3/20, to make a profit of $5000 with probability 9/20, to break even
with probability 1/4, and to lose $5000 with probability 3/20. The expected profit in
dollars is
A) 1500 B) 0 C) 3000 D) 3250 E) –1500
21. Refer to the previous problem. The standard deviation of profit would be about
A) 0 B) 1500 C) 2359 D) 4583 E) 21000000
22. A rock concert producer has scheduled an outdoor concert. If it is warm that day, she
expects to make a $20000 profit. If it is cool that day, she expects to make a $5000
profit. If it is very cold that day, she expects a $12000 loss. Based on historical records,
the weather office has estimated the chances of a warm day to be 0.40; the chances of a
cool day to be 0.5. What is the producer’s expected profit?
A) $9300 B) $9900 C) $10000 D) $105000 E) $11450
23. Suppose X is a continuous random variable taking values between 0 and 2 and having
the probability density curve below.
P(1 ≤ X ≤ 2) has value
A) 0.75 B) 0.5 C) 0.33
D) 0.25
E) 0
24. A random variable Y has the following distribution:
Y
P(Y)
–1
3C
0
2C
1
0.4
The value of the constant C is:
A) 0.1
B) 0.15 C) 0.2 D) 0.25
2
0.1
E) 0.75
25. A random variable X has a probability distribution as follows:
X
P(X)
0
2k
1
3k
2
13k
3
2k
The probability that P(X < 2.0) is equal to
A) 0.9
B) 0.25 C) 0.65 D) 0.15 E) 1
26. Cans of soft drinks cost $0.30 in a certain vending machine. What is the expected value
of daily revenue (Y) from the machine, if X, the number of cans sold per day has E(X) =
125?
A) 30 B) 37.5 C) 62.5 D) 107.5 E) 416.67
27. Refer to the previous problem. What is the variance of daily revenue (Y) from the
machine, if Var(X) = 50?
A) 50 B) 30
C) 11.25 D)7.07
E) 4.5
28. In a particular game, a fair die is tossed. If the number of spots showing is either a 4 or
5, you win $1. If the number of spots showing is 6, you win $4. If the number of spots
showing is 1, 2, or 3, you win nothing. Let X be the amount that you win. The expected
value of X is
A) $0 B) $1 C) $2.50 D) $4 E) $6
29. Suppose X is a random variable with mean X and standard deviation X. Suppose Y is a
random variable with mean Y and standard deviation Y. The mean of X + Y is
A) X + Y.
B) (X/X) + (Y/Y).
C) X + Y, but only if X and Y are independent.
D) (X/X) + (Y/Y), but only if X and Y are independent.
E) None of these.
30. Suppose X is a random variable with mean X and standard deviation X. Suppose Y is a
random variable with mean Y and standard deviation Y. The variance of X + Y is
A) X + Y.
B) (X)2 + (Y)2.
C) X + Y, but only if X and Y are independent.
D) (X)2 + (Y)2, but only if X and Y are independent.
E) None of these.
31. Suppose X is a continuous random variable taking values between 0 and 2 and having
the probability density function below.
P(0 ≤ X ≤ 2) has value
A) 0.5 B) 0.33 C) 0.25
D) 0
E) none of these
32. A fifth-grade teacher gives homework every night in both mathematics and language
arts. The time to complete the mathematics homework has a mean of 25 minutes and a
standard deviation of 8 minutes. The time to complete the language arts assignment has
a mean of 15 minutes and a standard deviation of 12 minutes. The time to complete the
mathematics homework and the time to complete the language arts homework have a
correlation of  = –0.3. The mean time to complete the entire homework assignment
A) is less than 40 minutes since the negative correlation tells you that more time on one
assignment will be associated with less time on the second assignment.
B) is 40 minutes.
C) is greater than 40 minutes since the measurements are correlated, which raises the
mean regardless of the sign of the correlation.
D) cannot be determined unless the times have a normal distribution.
E) is 60 minutes.
33. A seventh-grade teacher gives homework every night in both mathematics and language
arts. The time to complete the mathematics homework has a mean of 10 minutes and a
standard deviation of 3 minutes. The time to complete the language arts assignment has
a mean of 12 minutes and a standard deviation of 4 minutes. Assuming the times to
complete homework assignments in math and language arts are independent, the
standard deviation of the time required to complete the entire homework assignment is
A) 25 minutes. B) 10 minutes. C) 5 minutes. D) 1 minute. E) none of these.
34. An insurance company has estimated the following cost probabilities for the next year
on a particular model of car:
Cost
Probability
$0
0.60
$500
0.05
$1000
0.13
$2000
?
The expected cost to the insurance company is (approximately)
A) $155 B) $595 C) $875 D) $645 E) $495
35. Suppose we have a loaded die that gives the outcomes 1 to 6 according to the
probability distribution
X
P(X)
1
0.1
2
0.2
3
0.3
4
0.2
5
0.1
6
0.1
Note that for this die all outcomes are not equally likely, as would be if this die were
fair. If this die is rolled 6000 times, then X , the sample mean of the number of spots on
the 6000 rolls, should be about
A) 3 B) 3.3 C) 3.5 D) 4.5 E) 3.25
Use the following to answer questions 36-39:
The weight of medium-sized oranges selected at random from a bin at the local supermarket is a
normal random variable with mean  = 10 ounces and standard deviation  = 1 ounce.
36. Suppose we pick two oranges at random from the bin, so the weights of the oranges are
independent. The difference in the weights of the two oranges selected (the weight of
first minus the weight of the second) has which of the following distributions?
A) N(0, 0.5) B) N(0, 4) C) N(0, 2) D) N(0, 1.414) E) Uniform with mean 0
37. Referring to the information above, find the (approximate) probability that a randomly
selected orange will weigh less than 9 ounces.
A) 0.9999 B) 0.8413 C) 0.5 D) 0.3333 E) 0.1587
38. Referring to the information above, find the (approximate) probability that a randomly
selected orange will weigh more than 6 ounces.
A) 0.9999 B) 0.8413 C) 0.5 D) 0.3333 E) 0.1587
39. Referring to the information above, find the (approximate) probability that a randomly
selected orange will weigh less than 8.5 ounces.
A) 0.9999 B) 0.9332 C) 0.5 D) 0.0668 E) 0
40. A random variable X has mean X and standard deviation X. Suppose n independent
observations of X are taken and the average X of these n observations is computed. If n
is very large, the law of large numbers implies
A) that X will be close to X.
B) that X will be approximately normally distributed.
C) that the standard deviation of X will be close to X.
D) that X will be uniformly distributed.
E) all of the above.
Use the following to answer questions 41 and 42:
The temperature at any random location in a kiln used in the manufacture of bricks is normally
distributed with a mean of 1000o F and a standard deviation of 50° F.
41. If bricks are fired at a temperature above 1125°F, they will crack and must be discarded.
If the bricks are placed randomly throughout the kiln, the proportion of bricks that crack
during the firing process is closest to
A) 0.62% B) 2.28% C) 6.2% D) 47.72% E) 49.38%
42. When glazed bricks are put in the oven, they will miscolor if the temperature is below
900°F. If the bricks are placed randomly throughout the kiln, the proportion of glazed
bricks that miscolor is closest to
A) 0.62% B) 2.28% C) 22.8% D) 47.72% E) 49.38%
43. Birthweights at a local hospital have a normal distribution with a mean of 110 ounces
and a standard deviation of 15 ounces. What percent of infants will have birthweights
under 95 ounces?
A) 0.159 B) 0.341 C) 0.500 D) 0.682 E) 0.841
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